Integers: The Complete Number System: quick revision, 15 bites

Integers: Whole Numbers and Their Opposites

Zero sits at the center of everything.

Integers are all whole numbers and their negatives, including zero. They extend infinitely in both directions on the number line.

Takeaway. Integers = {..., -3, -2, -1, 0, 1, 2, 3, ...} — the complete set of whole numbers and their opposites.

What Exactly Is an Integer?

No fractions, no decimals — just whole numbers.

Takeaway. If it has a decimal or fraction, it's not an integer.

The Number Line: Integers in Order

See how integers line up forever.

The number line places integers in order: negatives to the left of zero, positives to the right. The farther right, the greater the value.

Takeaway. On the number line, -3 < -1 < 0 < 2 < 5 — left is always less.

Zero Is Special: Neither Positive Nor Negative

Zero breaks the positive/negative rule.

Zero is an integer, but it is **neither positive nor negative**. It is its own opposite and the boundary between positive and negative integers.

Takeaway. Zero is the only integer that is neither positive nor negative.

Absolute Value: Distance from Zero

Distance is always positive.

Takeaway. Absolute value gives the distance from zero — always non-negative.

Adding Integers: Same Signs vs Different Signs

Two rules cover every addition.

Takeaway. Same signs: add and keep sign. Different signs: subtract and take the sign of the larger absolute value.

Subtracting Integers: Add the Opposite

Subtraction is just addition in disguise.

To subtract an integer, **add its opposite**. This turns every subtraction into an addition problem.

Takeaway. a − b = a + (−b) — flip the sign and add.

Multiplying and Dividing Integers: Sign Rules

Same signs give positive, different give negative.

The same sign rules apply to division.

Takeaway. Same signs → positive; different signs → negative. Works for both × and ÷.

Don't Mix Up Addition and Multiplication Rules

The sign rules are NOT the same.

Takeaway. Addition and multiplication follow different sign rules — never mix them up.

Integers in Everyday Life

Temperature, money, and elevation use integers.

Takeaway. Integers model any quantity with direction — above/below, gain/loss, up/down.

Remember Sign Rules with a Simple Rhyme

Friends and enemies make signs easy.

Takeaway. Same signs = friends = positive; different signs = enemies = negative.

Exam Tip: Watch the Signs

Most integer errors are sign errors.

In exams, integer questions often test sign handling. Double-check every sign, especially when subtracting negatives or multiplying/dividing with mixed signs.

Takeaway. Slow down on signs — they are where marks are lost.

Using Integers to Solve Real Problems

Turn words into integer expressions.

Takeaway. Translate real situations into integer operations to solve them.

Integers vs. Whole Numbers vs. Natural Numbers

Know the difference between number sets.

Natural numbers are {1, 2, 3, ...} — they exclude zero as well.

Takeaway. Integers add negatives to whole numbers; whole numbers add zero to natural numbers.

The Word 'Integer' Comes from Latin

Integer means 'whole' in Latin.

The word **integer** comes from the Latin *integer*, meaning 'whole' or 'intact' — reflecting that integers are whole numbers.

Takeaway. Integer = whole — no fractions allowed.